Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces (original) (raw)
Related papers
2019
In weighted Orlicz type spaces S p, μ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is in a certain sense the best. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces S p, μ .
Direct and inverse approximation theorems in the weighted Orlicz-type spaceswith a variable exponent
TURKISH JOURNAL OF MATHEMATICS
In weighted Orlicz-type spaces S p, µ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre K-functionals is shown in the spaces S p, µ .
Approximation and moduli of fractional order in Smirnov-Orlicz classes
Glasnik Matematicki, 2008
In this work we investigate the approximation problems in the Smirnov-Orlicz spaces in terms of the fractional modulus of smoothness. We prove the direct and inverse theorems in these spaces and obtain a constructive descriptions of the Lipschitz classes of functions defined by the fractional order modulus of smoothness, in particular.
Smoothness of the Orlicz norm in Musielak-Orlicz function spaces
Mathematische Nachrichten, 2013
In this paper, we present a characterization of support functionals and smooth points in L Φ 0 , the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of L Φ 0 is also obtained. Some expressions involving the norms of functionals in (L Φ 0) * , the topological dual of L Φ 0 , are proved for arbitrary Musielak-Orlicz functions.
On moduli of smoothness and K-functionals of fractional order in the Hardy spaces
Journal of Mathematical Sciences, 2012
We prove the equivalence of special moduli of smoothness and K-functionals of fractional order in the space Hp, p > 0. As applications, we obtain an analog of the Hardy-Littlewood theorem and the sharp estimates of the rate of approximation of functions by generalized Bochner-Riesz means.
On approximation in Weighted Orlicz spaces
Mathematica Slovaca, 2012
An inverse theorem of the trigonometric approximation theory in Weighted Orlicz spaces is proved and the constructive characterization of the generalized Lipschitz classes defined in these spaces is obtained. c 2012 Mathematical Institute Slovak Academy of Sciences 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 41A25, 41A27, 42A10, 46E30. K e y w o r d s: best approximation, modulus of smoothness, Muckenhoupt weight, Orlicz space.
Gossez's approximation theorems in Musielak–Orlicz–Sobolev spaces
Journal of Functional Analysis, 2018
We prove the density of smooth functions in the modular topology in the Musielak-Orlicz-Sobolev spaces essentially extending the results of Gossez [16] obtained in the Orlicz-Sobolev setting. We impose new systematic regularity assumption on M which allows to study the problem of density unifying and improving the known results in the Orlicz-Sobolev spaces, as well as the variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of W 1,p 0 (Ω) functions by smooth functions in the double-phase space governed by the modular function H(x, s) = s p + a(x)s q with a ∈ C 0,α (Ω) excluding the Lavrentiev phenomenon within the sharp range q/p ≤ 1 + α/N. See [10, Theorem 4.1] for the sharpness of the result.
An approximation theorem in Musielak-Orlicz-Sobolev spaces
In this paper we prove the uniform boundedness of the operators of convolution in the Musielak-Orlicz spaces, and the density of D (R^n) in the Musielak-Orlicz-Sobolev spaces by assuming a condition of Log-Hölder type of continuity.